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A Person-Fit Index for Polytomous Rasch Models, Latent Class Models, and Their Mixture Generalizations

Published online by Cambridge University Press:  01 January 2025

Matthias von Davier  and
Ivo W. Molenaar
Matthias von Davier*
Affiliation:
Educational Testing Service
Ivo W. Molenaar
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Matthias von Davier, ETS, MS O2-R, Princeton NJ 08541. E-Mail: mvondavier@ets.org
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Abstract

A normally distributed person-fit index is proposed for detecting aberrant response patterns in latent class models and mixture distribution IRT models for dichotomous and polytomous data.

This article extends previous work on the null distribution of person-fit indices for the dichotomous Rasch model to a number of models for categorical data. A comparison of two different approaches to handle the skewness of the person-fit index distribution is included.

Keywords

person-fitpolytomous Rasch modelmixed Rasch modellatent class analysisappropriateness measurement
Type
Articles
Information
Psychometrika , Volume 68 , Issue 2 , June 2003 , pp. 213 - 228
Copyright
Copyright © 2003 The Psychometric Society

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Footnotes

Major parts of this paper were written while the first author worked at the Institute for Science Education, Kiel, Germany. Any opinions expressed in this paper are those of the authors and not necessarily of Educational Testing Service. The results presented in this paper were improved by valuable comments from J. Rost, K. Yamamoto, N.D. Verhelst, E. Bedrick and two anonymous reviewers.

References

Aitkin, M., Anderson, D., Hinde, J. (1981). Statistical modeling of data on teaching style (with discussion). Journal of the Royal Statistical Society, 144, 419–461.CrossRefGoogle Scholar
Andrich, D. (1982). An extension of the Rasch model for ratings providing both location and dispersion parameters. Psychometrika, 47, 105–113.CrossRefGoogle Scholar
Bedrick, E. J. (1997). Approximating the conditional distribution of person fit indexes for checking the Rasch model. Psychometrika, 62, 191–199.CrossRefGoogle Scholar
Biernacki, C., Celeux, G., Govaert, G. (2000). Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Transactions on pattern analysis and machine intelligence, 22, 719–725.CrossRefGoogle Scholar
Celeux, G., Soromenho, G. (1996). An Entropy Criterion for Assessing the Number of Clusters in a Mixture Model. Journal of Classification, 13, 195–212.CrossRefGoogle Scholar
Drasgow, F., Levine, M. V., Williams, E. A. (1985). Appropriateness measurement with polychotomous item response models and standardized indices. British Journal of Mathematical and Statistical Psychology, 38, 67–86.CrossRefGoogle Scholar
Kendall, M., Stuart, A. (1967). The advanced theory of statistics, Vol. 1. New York, NY: Macmillan.Google Scholar
Langeheine, R., Pannekoek, J., van de Pol, F. (1996). Bootstrapping goodness-of-fit measures in categorical data analysis. Sociological Methods and Research, 24, 492–516.CrossRefGoogle Scholar
Lazarsfeld, P.F., Henry, N.W. (1968). Latent structure analysis. Boston, MA: Houghton Miffin.Google Scholar
Liou, M. (1993). Exact person tests for assessing model-data fit in the Rasch model. Applied Psychological Measurement, 17, 187–195.Google Scholar
Liou, M., Chang, C.H. (1992). Constructing the exact significance level for a person-fit statistic. Psychometrika, 57, 169–181.CrossRefGoogle Scholar
Masters, G. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 147–174.CrossRefGoogle Scholar
McCullagh, P. (1986). The conditional distribution of goodness-of-fit statistics for discrete data. Journal of the American Statistical Association, 81, 104–107.CrossRefGoogle Scholar
McLachlan, G.J., Peel, D. (1997). On a resampling approach to choosing the number of components in normal mixture models. In Billard, L., Fisher, N.I. (Eds.), Computing Science and Statistics (pp. 260–266). Fairfax Station, VA: Interface Foundation of North America.Google Scholar
Molenaar, I.W. (1983). Some improved diagnostics for failure of the Rasch model. Psychometrika, 48, 49–72.CrossRefGoogle Scholar
Molenaar, I.W., Hoijtink, H. (1990). The many null distributions of person fit indices. Psychometrika, 55, 75–106.CrossRefGoogle Scholar
Molenaar, I.W., Hoijtink, H. (1996). Person-fit and the Rasch model, with an application to knowledge of logical quantors. Applied Measurement in Education, 9, 27–45.CrossRefGoogle Scholar
Rost, J. (1990). Rasch models in latent classes: An integration of two approaches to item analysis. Applied Psychological Measurement, 14, 271–282.CrossRefGoogle Scholar
von Davier, M. (1997). Methoden zur Prüfung probabilistischer Testmodelle [Methods for checking probabilistic test models], Kiel, Germany: IPN.Google Scholar
von Davier, M. (1997b). Bootstrapping goodness-of-fit statistics for sparse categorical data: Results of a Monte Carlo study. Methods of Psychological Research. Available online at http://www.mpr-online.de/ [Click on “Currently available issues”. Then select “1997 Vol. 2 No. 2”.]Google Scholar
von Davier, M. (2000). Winmira 2001—A Microsoft WindowsTM program for analyses with the Rasch model, with the latent class analysis and with the mixed Rasch model [Computer Software]. Available for download from http://winmira.von-davier.deGoogle Scholar
von Davier, M., Rost, J. (1995). Mixture distribution Rasch models. In Fischer, G.H., Molenaar, I.W. (Eds.), Rasch models—Foundations, recent developments, and applications (pp. 257–268). New York, NY: Springer.Google Scholar
von Davier, M., Rost, J. (1995). Polytomous mixed Rasch models. In Fischer, G. H., Molenaar, I. W. (Eds.), Rasch models—Foundations, recent developments, and applications (pp. 371–379). New York, NY: Springer.Google Scholar
Windham, M.P. (1983). Geometrical Fuzzy Clustering Algorithms. Fuzzy Sets and Systems, 10, 271–279.CrossRefGoogle Scholar
Yamamoto, K. (1987). A model that combines IRT and latent class models. Unpublished doctoral dissertation, University of Illinois.Google Scholar